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Alan Turing was a complex man doing insanely complex work, and in “The Imitation Game,” he’s played by Benedict Cumberbatch, an actor born for this kind of role.

Cumberbatch nails it. If only the film did, too.

Turing was a brilliant British mathematician and cryptologist who helped crack the Enigma code during World War II, a code used by the Nazis that was previously considered unbreakable. He more or less invented, or at least thought up, the idea of a computer. He was a secret hero — those working on the Enigma project are told up front if they tell anyone what they’re doing, they’ll be hanged for treason. The work depicted here remained unknown until the 1990s.

Turing also was a casualty of bigotry, prosecuted in the 1950s for homosexual acts then considered illegal in England. Made to choose between prison and a form of chemical castration, he chose the latter, before committing suicide at 41.

But director Morten Tyldum, working from a screenplay by Graham Moore (based on the book by Andrew Hodges), takes a surprisingly straightforward approach to the story. Everything fits together too neatly, with far too few rough edges and surprises.

A framing device finds Turing being questioned by a detective (Rory Kinnear), who doesn’t care about his sex life — he thinks Turing is a spy. Turing tells the drop-jawed detective his story, about how he and other mathematicians were recruited to work at Bletchley Park, where they feverishly attempt to crack Enigma. He butts heads with Commander Denniston (Charles Dance), the old-school leader at Bletchley, and with Hugh Alexander (Matthew Goode), who was running the Enigma project until, with the help of Winston Churchill (!), Turing replaces him.

Turing thinks the solution can be found in a machine he builds and constantly tinkers with. The rest of the team thinks he’s crazy. You can guess where this is leading, just like you can guess most of what’s going to happen here.

Joan Clarke (Keira Knightley) joins the team, once she gets past the prejudices against women being smart enough for this kind of work. She and Turing bond, and a platonic relationship is hatched (as is a proposal for marriage). And, slowly but surely, the team comes around to respecting the difficult, tactless Turing, leading to a kind of “I am Spartacus!” moment when Denniston tries to shut down the machine because it hasn’t yielded any results.

Give it time, man. Then again, time is something the Allies don’t have much of.

Tyldum goes back and forth between the time at Bletchley and Turing’s childhood (the young Turing is played by Alex Lawther, also good), where he is “different” at boarding school — just kind of arrogant, not well-liked because he is smarter than everyone else and isn’t shy about saying so. He acts similarly at Bletchley.

Turing develops a deep friendship with an older boy that turns to love, though chaste; this will inform, poignantly, everything in the rest of Turing’s life.

But the film can’t help itself. Even after the code is cracked — it’s not exactly a spoiler; we did win the war, after all, and Churchill gave Turing and his team a large amount of credit — Moore’s script makes another immediate leap into melodrama. Why? There is invention, brilliance and tragedy aplenty here without it.

But when it’s good, “The Imitation Game” is very good. Cumberbatch is terrific, which is not surprising, given the marriage of role and actor. Goode, Dance and Knightley also are quite good, as is Mark Strong as an MI6 boss who understands right away Turing’s social challenges — and his genius. He, like Clarke, recognizes the complexity of Turing’s life. Too bad the movie too often doesn’t.

REVIEW

“THE IMITATION GAME”

★★★

Benedict Cumberbatch is great as Alan Turing, a British mathematician who helped break a Nazi code and was later prosecuted for being gay. But the film itself is far less complex than his performance. PG-13. 114 minutes.